Multiple shooting shadowing for sensitivity analysis of chaotic dynamical systems

Blonigan, Patrick; Wang, Qiqi

doi:10.1016/j.jcp.2017.10.032

Cited by 36 publications

(60 citation statements)

References 69 publications

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“…As already argued in Ref. [6] and references therein, the hope lies in the so called "chaotic hypothesis" [14]. The hypothesis is that structural changes in the attractor as parameters are varied are not that catastrophic if the dimension of the system is large enough, in the "thermodynamic limit" [32].…”

Section: Discussion On the Bias Errormentioning

confidence: 99%

Periodic shadowing sensitivity analysis of chaotic systems

Lasagna

Sharma

Meyers

2019

Journal of Computational Physics

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The sensitivity of long-time averages of a hyperbolic chaotic system to parameter perturbations can be determined using the shadowing direction, the uniformly-bounded-in-time solution of the sensitivity equations. Although its existence is formally guaranteed for certain systems, methods to determine it are hardly available. One practical approach is the Least-Squares Shadowing (LSS) algorithm (Q Wang, SIAM J Numer Anal 52, 156, 2014), whereby the shadowing direction is approximated by the solution of the sensitivity equations with the least square average norm. Here, we present an alternative, potentially simpler shadowing-based algorithm, termed periodic shadowing. The key idea is to obtain a bounded solution of the sensitivity equations by complementing it with periodic boundary conditions in time. We show that this is not only justifiable when the reference trajectory is itself periodic, but also possible and effective for chaotic trajectories. Our error analysis shows that periodic shadowing has the same convergence rates as LSS when the time span T is increased: the sensitivity error first decays as 1{T and then, asymptotically as 1{?T . We demonstrate the approach on the Lorenz equations, and also show that, as T tends to infinity, periodic shadowing sensitivities converge to the same value obtained from long unstable periodic orbits (D Lasagna, SIAM J Appl Dyn Syst 17, 1, 2018) for which there is no shadowing error. Finally, finite-difference approximations of the sensitivity are also examined, and we show that subtle non-hyperbolicity features of the Lorenz system introduce a small, yet systematic, bias.

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Section: Discussion On the Bias Errormentioning

confidence: 99%

Periodic shadowing sensitivity analysis of chaotic systems

Lasagna

Sharma

Meyers

2019

Journal of Computational Physics

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“…The shadowing trajectory is guaranteed to exist for uniformly hyperbolic systems [32]. Blonigan and Wang [28] introduced the MSS algorithm, which is a variant of LSS. For the two trajectories to remain close to each other, the reference trajectory is split into K segments and the following minimization problem is solved:…”

Section: B Computation Of Sensitivities Using Mssmentioning

confidence: 99%

“…(12) for w, the QoI J = dG ∞ (s) ds can be easily computed (refer to Refs. [28,34]). We are now ready to couple the two algorithms to predict the statistical behavior of J.…”

Section: B Computation Of Sensitivities Using Mssmentioning

confidence: 99%

“…This is an important restriction because, in the area of fluid mechanics for example, most engineering flows become chaotic beyond a certain Reynolds number. In a series of papers, Wang and co-workers have introduced the idea of least-squares shadowing (LSS) to address this problem [24][25][26][27][28]. Other approaches have also been proposed [29,30].…”

Section: Introductionmentioning

confidence: 99%

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Uncertainty quantification of sensitivities of time-average quantities in chaotic systems

2020

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We consider time-average quantities of chaotic systems and their sensitivity to system parameters. When the parameters are random variables with a prescribed probability density function, the sensitivities are also random. The central aim of the paper is to study and quantify the uncertainty of the sensitivities; this is useful to know in robust design applications. To this end, we couple the nonintrusive polynomial chaos expansion (PCE) with the multiple shooting shadowing (MSS) method, and apply the coupled method to two standard chaotic systems, the Lorenz system and the Kuramoto-Sivashinsky equation. The method leads to accurate results that match well with Monte Carlo simulations (even for low chaos orders, at least for the two systems examined), but it is costly. However, if we apply the concept of shadowing to the system trajectories evaluated at the quadrature integration points of PCE, then the resulting regularization can lead to significant computational savings. We call the new method shadowed PCE (sPCE).

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“…Although taking snapshots does not reduce the computational complexity, it reduces data storage. The idea of taking snapshots was also used in the multiple-shooting shadowing method developed by Blonigan [37]. The large part of the FD-NILSS algorithm is to compute {a i } K−1 i=0 , using which we can construct the shadowing direction as shown in [32]: this does not use any knowledge of the instantaneous objective function J(u, s).…”

Section: Computementioning

confidence: 99%

Sensitivity analysis on chaotic dynamical systems by Finite Difference Non-Intrusive Least Squares Shadowing (FD-NILSS)

Wang

Fernández

et al. 2019

Journal of Computational Physics

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We present the Finite Difference Non-Intrusive Least Squares Shadowing (FD-NILSS) algorithm for computing sensitivities of long-time averaged quantities in chaotic dynamical systems. FD-NILSS does not require tangent solvers, and can be implemented with little modification to existing numerical simulation software. We also give a formula for solving the least-squares problem in FD-NILSS, which can be applied in NILSS as well. Finally, we apply FD-NILSS for sensitivity analysis of a chaotic flow over a 3-D cylinder at Reynolds number 525, where FD-NILSS computes accurate sensitivities and the computational cost is in the same order as the numerical simulation.

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Multiple shooting shadowing for sensitivity analysis of chaotic dynamical systems

Cited by 36 publications

References 69 publications

Periodic shadowing sensitivity analysis of chaotic systems

Periodic shadowing sensitivity analysis of chaotic systems

Uncertainty quantification of sensitivities of time-average quantities in chaotic systems

Sensitivity analysis on chaotic dynamical systems by Finite Difference Non-Intrusive Least Squares Shadowing (FD-NILSS)

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